Revisiting Baer elements

TL;DR

This paper extends properties of $d$-ideals and $d$-elements to Baer elements in multiplicative lattices, introducing new concepts like Baer closures and analyzing their maximal, prime, and irreducible forms.

Contribution

It generalizes properties from ring theory to multiplicative lattices and introduces new Baer-related concepts and results not previously addressed.

Findings

Extended properties of $d$-ideals to Baer elements in lattices

Introduced Baer closures and analyzed their properties

Identified new results on Baer maximal, prime, and semiprime elements

Abstract

The objective of this paper is to extend certain properties observed in $d$-ideals of rings and $d$-elements of frames to Baer elements in multiplicative lattices introduced in D. D. Anderson, C. Jayaram, and P. A. Phiri, Baer lattices, \textit{Acta Sci. Math. (Szeged)}, 59 (1994), 61--74. Additionally, we present results concerning these elements that have not been addressed in the study of $d$-ideals of rings. Furthermore, we introduce Baer closures and explore Baer maximal, prime, semiprime, and meet-irreducible elements.